On the sum of partition norms and its connection to norms of partitions with parts greater than one

Meenakshi Rana, Harman Kaur and Abhimanyu Kumar
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 908–915
DOI: 10.7546/nntdm.2025.31.4.908-915
Full paper (PDF, 246 Kb)

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Authors and affiliations

Meenakshi Rana
School of Mathematics, Thapar Institute of Engineering and Technology
Patiala, Punjab 147004, India

Harman Kaur
Department of Mathematics, Chandigarh University
Mohali, Punjab 140413, India

Abhimanyu Kumar
Department of Electrical and Instrumentation Engineering, Thapar Institute of Engineering and Technology
Patiala, Punjab 147004, India

Abstract

We study the partition norm—the product of the parts of a partition—with emphasis on partitions whose parts all exceed 1. We obtain two bivariate recurrences for the sum of norms over partitions into distinct parts, including a refinement that separates the contributions of the part 1 and of the prime 2. To count partitions of with parts greater than 1 having norm , we introduce the restricted norm-counting function , give its two-parameter generating function, and derive recurrences and other relations. Finally, we formulate analogues of Goldbach’s and twin-prime conjectures in the language of partition norms.

Keywords

• Integer partitions
• Partition norm
• Generating functions
• Recurrences

2020 Mathematics Subject Classification

• 05A17
• 05A10
• 05A15
• 11N99
• 11P32

References

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Manuscript history

• Received: 1 April 2025
• Revised: 4 November 2025
• Accepted: 8 December 2025
• Online First: 10 December 2025

Copyright information

Ⓒ 2025 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).